Problem 52 - Precalculus Student Solutions Manual 5th

Question

Find each product. Write the answer in standard form. $$(-2+3 i)(4-2 i)$$

Step-by-Step Solution

Answer

-2 + 16i

Step 1: Use the distributive property

Apply the distributive property (also known as the FOIL method for binomials) to find the product \((-2+3i)(4-2i)\). Multiply each term in the first complex number by each term in the second complex number.

Step 2: Multiply the first terms

Multiply the real parts first: \((-2) \times 4 = -8\).

Step 3: Multiply the outer terms

Multiply the real part of the first number by the imaginary part of the second number: \((-2) \times (-2i) = 4i\).

Step 4: Multiply the inner terms

Multiply the imaginary part of the first number by the real part of the second number: \(3i \times 4 = 12i\).

Step 5: Multiply the last terms

Multiply the imaginary parts: \(3i \times (-2i) = -6i^2\). Since \(i^2 = -1\), this becomes: \(-6(-1) = 6\).

Step 6: Combine like terms

Combine all the results: \(-8 + 4i + 12i + 6\). Combine the real terms and the imaginary terms: \((-8 + 6) + (4i + 12i) = -2 + 16i\).

Step 7: Write the answer in standard form

The standard form of a complex number is \(a + bi\). Therefore, the result is \(-2 + 16i\).

Key Concepts

distributive property

The distributive property is a fundamental algebraic principle, crucial when working with complex numbers. It states that for any three numbers, say, A, B, and C, the following equation holds true: ewline ewline A(B + C) = AB + AC. ewline ewline When we extend this property to complex numbers, it allows us to break down multiplication into manageable parts. For instance, given the problem ewline ewline (-2+3i)(4-2i),ewlineewline we distribute each term in the first complex number by each term in the second complex number. This way, ewline ewline (-2+3i)(4-2i) = (-2)(4) + (-2)(-2i) + (3i)(4) + (3i)(-2i). ewline ewline This method simplifies multiplication, ensuring we handle each product separately before combining the results.

FOIL method

The FOIL method is a handy mnemonic for remembering how to multiply two binomials. FOIL stands for First, Outer, Inner, Last, indicating the pairs of terms we multiply to expand the product. In our example, ewline ewline (-2+3i)(4-2i),ewline ewline we apply the FOIL method as follows: ewline ewline

• First: Multiply the first terms in each binomial: ewline (-2)(4) = -8.
• Outer: Multiply the outermost terms: ewline (-2)(-2i) = 4i.
• Inner: Multiply the innermost terms: ewline (3i)(4) = 12i.
• Last: Multiply the last terms in each binomial: ewline (3i)(-2i) = -6i^2. Since i^2 = -1, this becomes -6(-1) = 6.

ewline ewline Finally, we combine all these products: ewline -8 + 4i + 12i + 6.

standard form of complex numbers

To represent complex numbers in a standardized and easily understandable form, we use the format ewline ewline a + bi, where ewline ewline 'a' is the real part and 'b' is the imaginary part. After performing all the necessary multiplications and combinations using the distributive property or the FOIL method, we need to combine like terms. Specifically: ewline ewline

• Combine all real parts together.
• Combine all imaginary parts together.

ewline In our example: ewline -8 + 4i + 12i + 6, combining the real parts, we get: ewline -8 + 6 = -2, andcombining the imaginary parts, we get: ewline 4i + 12i = 16i. Therefore, the result in standard form is ewline ewline -2 + 16i. This presentation is clear and precise, making it straightforward for anyone familiar with complex numbers to understand.